We show that Quillen’s small object argument works for exact categories
under very mild conditions. This has immediate applications to cotorsion
pairs and their relation to model structures, and to the existence of
certain triangulated adjoint functors. In particular, the interplay of
different exact structures on the category of complexes of
quasi-coherent sheaves leads to a streamlined and generalized version of
recent results obtained by Neeman, Murfet, Hovey and Gillespie.